Logic Colloquium 90. ASL Summer Meeting in Helsinki

Lecture Notes in Logic J. Oikkonen J. Vaananen (Eds.)

Logic Colloquium '90 ASL Summer Meeting in Helsinki

Springer-Verlag

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Lecture Notes in Logic Editors: K. Fine, Los Angeles J.-Y. Girard, Marseille A. Lachlan, Burnaby T. Slaman, Chicago H. Woodin, Berkeley

J. Oikkonen J. Vaananen (Eds.)

Logic Colloquium '90 ASL Summer Meeting in Helsinki

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Editors Juha Markku Robert Oikkonen Jouko Antero Vaananen Department of Mathematics P. O. Box 4 (Hallituskatu 15) SF-00014 University of Helsinki, Finland

Mathematics Subject Classification (1991): OOB20 ISBN 3-540-57094-2 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57094-2 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Printed in Germany 46/3140-543210 - Printed on acid-free paper

FOREWORD The 1990 European Summer Meeting of the Association for Symbolic Logic was held in Finland from July 15 to July 22, 1990. The meeting was called Logic Colloquium '90 and it took place in the Porthania building of the University of Helsinki as part of the program of the 350th anniversary of the university. The meeting was attended by 140 registered participants and 31 accompanying persons, from 23 different countries. The organizing bodies were the Department of Mathematics of the University of Helsinki, The Philosophical Society of Finland, and The Finnish Mathematical Society. Financial support was received from the Ministry of Education of Finland, The Academy of Finland, Suomen Kulttuurirahasto Foundation, UNESCO, Rolf Nevanlinna Institute, and IUHPS. The Organizing Committee of the meeting consisted of Aapo Halko, Heikki Heikkila, Lauri Hella, Taneli Huuskonen, Tapani Hyttinen, Kerkko Luosto, Ilkka Niiniluoto (vice chairman), Juha Oikkonen, and Jouko Vaananen (chairman), all from the University of Helsinki. The Program Committee consisted of Peter Aczel (Manchester), Max Dickmann (Paris), Heinz-Dieter Ebbinghaus (Freiburg), Jens Fenstad (Oslo), Jaakko Hintikka (chairman, Boston), Wilfrid Hodges (London), Alistair Lachlan (Vancouver), Azriel Levy (Jerusalem), Heikki Mannila (Helsinki), Ilkka Niiniluoto (Helsinki), Juha Oikkonen (Helsinki), and Jouko Vaananen (secretary, Helsinki). The program of the meeting is listed on the following pages. Warren Goldfarb, Ronald Jensen, Phokion Kolaitis, Per Martin-Lδf, Alan Mekler, and Hugh Woodin did not contribute a paper to the proceedings. As comparison between the contents of this book and the actual program reveals, some authors made an agreement with the editors to contribute a slightly different paper from the one read in the meeting. Also Joan Moschovakis and Alan Silver were approached by the editors and they submitted the paper they read in a contributed papers session of the meeting. The editors are indebted to Heikki Heikkila, Yiannis Moschovakis, Martti Nikunen, and Hannele Salminen for their help during the preparation of this volume. We owe special thanks to Herbert Enderton for the substantial work he has done in putting together the final manuscript. Juha Oikkonen Jouko Vaananen

Invited talks of Logic Colloquium '90 WlLFRIED BUCHHOLZ (Mϋnchen) Cut-elimination in uncountable logic and collapsing functions BARRY COOPER (Leeds) Definability and global degree theory PATRICK DEHORNOY (Caen) About the word problem for free left distributive groupoids HANS-DIETER BONDER (Mϋnchen) On ωi'Complete filters Dov GABBAY (London) Labelled deductive systems WARREN GOLDFARB (Harvard) On Gόdel's philosophy JAAKKO HINTIKKA (Boston) Is there completeness in mathematics after Go del? IAN HODKINSON (London) An axiomatisation of the temporal logic with until and since over real numbers RONALD JENSEN (Oxford) Remarks on the core model HAIM JUDAH (Bar-Han) Δ\-sets of reals PHOKION KOLAITIS (Santa Cruz) 1. Logical definability and complexity classes 2. Model theory of finite structures 3. 0-1 laws RICHARD LAYER (Boulder) Elementary embeddings of a rank into itself PER MARTIN-Lor (Stockholm) Logic and metaphysics ALAN MEKLER (Vancouver) Almost free algebras: 20 years of progress GRIGORI MINTS (Stanford) Gentzen-type systems and resolution rule for modal predicate logic YlANNlS MOSCHOVAKIS (Los Angeles) Sense and denotation as algorithm and value TULENDE MUSTAFIN (Karaganda) On similarities of complete theories

VIII LUDOMIR NEWELSKI (Wroclaw) Geometry of finite rank types FRANgoiSE POINT (Paris) Decidability problems for theories of modules JEAN-PIERRE RESSAYRE (Paris) Discrete subrings of real closed fields and applications to polynomial time computability SAHARON SHELAH (Jerusalem) Indiscernibility HUGH WOODIN (Berkeley) Large cardinals and descriptive set theory

CONTENTS WlLFRIED BUCHHOLZ

1

A note on the ordinal analysis of KPM STEVEN BUECHLER and LUDOMIR NEWELSKI On the geometry of U-rank 2 types

10

S. BARRY COOPER Definability and global degree theory

25

PATRICK DEHORNOY About the irreflexivity hypothesis for free left distributive magmas

. . . .

46

HANS-DIETER BONDER On ω\-complete filters

62

D. M. GABBAY Labelled deductive systems: a position paper

66

D. M. GABBAY, I. M. HODKINSON, and M. A. REYNOLDS Temporal expressive completeness in the presence of gaps

89

JAAKKO HINTIKKA New foundations for mathematical theories

122

HAIM JUDAH Absoluteness for projective sets

145

RICHARD LAYER A division algorithm for the free left distributive algebra

155

GRIGORI MINTS Gentzen-type systems and resolution rule. Part II. Predicate logic . . . . 163 JOAN RAND MOSCHOVAKIS An intuitionistic theory of lawlike, choice and lawless sequences

191

YIANNIS N. MOSCHOVAKIS Sense and denotation as algorithm and value

210

M. H. MOURGUES and J.-P. RESSAYRE A transfinite version of Puiseux's theorem, with applications to real closed fields

250

T. G. MUSTAFIN On similarities of complete theories

259

FRANQOISE POINT Decidability questions for theories of modules

266

SAHARON SHELAH 1 On CH + 2* -> (a)\ for a < ω2

281

ALAN P. SILVER On the structure of gamma degrees

290

A NOTE ON THE ORDINAL ANALYSIS OF KPM W. BUCHHOLZ l

This note extends our method from (Buchholz [2]) in such a way that it applies also to the rather strong theory KPM. This theory was introduced and analyzed proof-theoretically in (Rathjen [6]), where Rathjen establishes an upper bound for its proof theoretic ordinal |KPM|. The bound was given in terms of a primitive recursive system T(M) of ordinal notations based on certain ordinal functions χ , ψκ (ω < K < M, /c regular) 2 that had been introduced and studied in (Rathjen [5]). 3 In section 1 of this note we define and study a slightly different system of functions φκ (K < M)—where ψM plays the role of Rathjen's χ—that is particularly well suited for our purpose of extending [2]. In section 2 we describe how one obtains, by a suitable modification of [2], an upper bound for |KPM| in terms of the φ^s from section 1. We conjecture that this bound is best possible and coincides with the bound given in [6]. In section 3 we prove some additional properties of the functions φκ which are needed to set up a primitive recursive ordinal notation system of ordertype > tf*, where ΰ* := Vtoi^M+i *s the upper bound for |KPM| determined in section 2. Remark: Another ordinal analysis of KPM has been obtained independently by T. Arai in Proof theory for reflecting ordinals II: recursively Mahlo ordinals (handwritten notes, 1989). §1. Basic properties of the functions ψκ (K < M). Preliminaries. The letters α, /9,7, ί, μ,σ, £,r/,C always denote ordinals. On denotes the class of all ordinals, and Lim the class of all limit numbers. Every ordinal a is identified with the set {£ G On : £ < a} of its predecessors. For a < β we set [α, β[ := {ξ : a < ξ < β}. By H- we denote ordinary (noncommutative) ordinal addition. An ordinal a > 0 which is closed under + is called an additive principal number. The class of all additive principal numbers is denoted by AP. The Veblen function φ is defined by φaβ := φa(β], where φa is the ordering function of the class {β 6 AP : V£ < a(φ^(β) — β)}. An ordinal 7 > 0 which is closed under φ (and thus also under +) is said to be strongly critical. The class of all strongly critical ordinals is denoted by SC. J

The final version of this paper was written while the author was visiting Carnegie Mellon University during the academic year 1990/91. I would like to express my sincere thanks to Wilfried Sieg (who invited me) and all members of the Philosophy Department of CMU for their generous hospitality. 2 M denotes the first weakly Mahlo cardinal. 3 The essential new feature of [5] is the function χ, while the ψκ's (K < M) are obtained by a straightforward generalization of previous constructions in [1], [3], [4].

2

W. BUCHHOLZ

Some basic facts: 1. AP = {ω > 7n sucn that 7 = 7o H ----- \~ Ίn4. For each 7 G AP \ SC there are uniquely determined £, η < 7 such that 7 = 1 and 7o > ••• > 7n are additive principal numbers. 4. SC(φtη) := SCtf) U SCfo), if £,•/ < ^ij. P7e assume the existence of a weakly Mahlo cardinal M. So every closed unbounded (club) set X C M contains at least one regular cardinal, and M itself is a regular cardinal. DEFINITION 1.1. R : = { α : u > < α < M fe α regular} MΓ := min{7 € SC : M < 7} = closure of M U {M} under +, φ SCu(Ί):=SC(Ί)nU Ω0 := 0 , Ωσ := K σ for σ > 0. Ω := the function σ *-+ ίlσ restricted to σ < M Remark: VAC € R( AC = ί7κ or AC e {Ωσ+1 : σ < M} ) Convention, /n £Λe following the letters AC, TT, r always denote elements of R. DEFINITION 1.2 (The collapsing functions ψκ). By transfinite recursion on a we define ordinals ψκa and sets C(a,β) C On as follows. Under the induction hypothesis that ψπξ and C(£, η) are already defined for all ξ < a , TT E R , η 6 On we set 1. C(α, /3) := closure of β U {0, M} under +, φ, Ω, ψ\a, where ψ\a denotes the binary function given by := { ( π , ξ ) :ξ κα £ R C;0Λα€SC\{Ωσ:σ< d) AC G C(α, /c) 7 € C» => 7 G CM(7) & 5CM(7) = SC(Ί) \ {M} 7 < * & 7 € C(α,β) = Proof. a),b) 1. Cκ(α) Π /c = ψκα is a trivial consequence of the definition of ψκα. 2. Let K — M. Obviously there exists a δ < K such that R Π [δ, κ[C T>κ(α). Therefore in order to get ψκα < /c it suffices to prove that the set U := {β G /c : C(α, β) Π AC C /?} is closed unbounded (club) in /c. i) c/oserf: Let 0 ^ X C {/ and β := sup(X) < AC. Then C(α,β) Π /c = U € ^(C'(α, 0 Π /c) C U^x ί = /?, i.e. βeU. ii) unbounded: Let /?0 < /c. We define /?n+1 := min{τ/ : C(α,βn) Γ\ K C η} and /? := sup n<w/ S n . Using L.l.lc we obtain βn < /?n+1 < AC. Hence β0 < β < K and C(α,/3) Π AC = Unκα < β < AC. — Now assume that ^>κα G R. We prove βn < ψκα (Vn). By definition of β0 and by L.I. la we have β0 < ψκα & /?0 ^ Lim. Hence βQ < ψκα. From βn < ψκα G R it follows that C(α,βn) Π AC C φκα and card((7(α,/?n) Π AC) < ^/>κα, and therefore βn^ < ψκα. From Vn(/3n < 0/βα ^ R) we βet /^ < ^κα Contradiction. c) 1. Obviously Cκ(α) Π AC is closed under φ. Together with a) this implies ψκa G SC. — 2. We have (ψκa = Ωσ > σ =Φ ^«« € ^(α)) and (by a) ) V> Λ α i Cκ(a). Hence ^κα ^ {Ωσ : σ < Ω σ }. d) follows from L.I. la, L.1.3a and the definition of ψκa. e) By L.1.3a VπGR(^πί < M) and therefore (7(α, M) = MΓ. As in d) one obtains f) and g) follow from e).

LEMMA 1.4. a) 7 € C(α,β) κα < Ωσ+1 . Then we have σ + 1 < AC and thus CΛ(α) Π AC = φκa < Ωσ+1 < Ωκ = /c. This implies Ωσ+α £ CΛ(α) and then (by a),b) ) σ ^ CΛ(α). Hence ^κα < σ < Ωσ < Y>κα. LEMMA 1.5. a) αQ < α & α0 G CM(α) =» ι/>Mα0 < Proof. a) From the premise we get ^Mαo ^ ^fM(α) Π M = ^Mα ^Y L.1.3a,g. b) Assume ^Mαo = ^M**! ^ αo < αι < ^Γ Then α0 G CM(αQ) C C^α^ and therefore by a) V>Mαo < ^Mαι Contradiction. LEMMA 1.6. For K < M the following holds a) aQ φκα0 < ψκα

b) α0κα0 < ^κ« Froo/. a) From α0 < α it follows that C(α0, V>*α) Π « C ^Ka- By definition of ψκαQ it therefore suffices to prove ψκα G {/9 : AC G C(o:0, AC) => AC G (^(αo,^)}. So let AC G C^αg, AC). — We have to prove AC G C(αQ,ψκα). CASE 1: AC = Ωσ+1. By Lemma 1.4c we have Ωσ < ψκα and therefore σ -f 1 G (7(0:0, ψκθi) which implies AC G C(αQ,ψκα). CASE 2: AC = Ω κ . From AC G C(α 0 ,Ac) C (7(α, AC) we obtain AC G ^(^o) Π Cκ(α). From this by L.1.2, L.1.3b, L.1.5b it follows that AC = φ^ξ with £ < α0 and ξ G (7Λ(α). Now by L.1.4a, L.1.3a,e we get 5CM(0 C Cκ(α) Π CM(ί) Π M = Cκ(α) Π AC = 0κα, and then ξ G (7(α0, V^) (by L.1.3f). From this together with ξ < α0 we obtain AC = V>M£ G C(α0,φκα) (by L.1.3g). b) The premise together with a) implies α0 < α fc Ac,α 0 G Cκ(α) Π Cκ(αQ) which gives us φκα0 G Cκ(α) Π AC = ^α. DEFINITION 1.3. For each set X C On we set W 7 (X) := n{C(a,^) : X C C(α,/?) & 7 < α}. §2. Ordinal analysis of KPM. In this section we show how one has to modify (and extend) [2] in order to establish that the ordinal Vto^M+i ιs an upper bound for |KPM|. Of course we now assume that the reader is familiar with [2]. The theory KPM is obtained from KPi by adding the following axiom scheme: (Mahlo) Vx3yφ(x, y, z ) -* 3w(Ad(w] Λ Vx€w3y€wφ(x, y, z)\

(φ G Δ0)

We extend the infinitary system RS°° introduced in Section 3 of [2] by adding the following inference rule:

ORDINAL ANALYSIS OF KPM

f\κ IΛ

(Mah)

>

,

-_

-— Γ, 3w€lM(Ad(w) Λ B(w)) : α

x (α 0 + M < α)

where B(w) is of the form Vxew3yewA(x,y) We set R := {a : ω < a < M & a regular}.

with k(Λ) C M.

Then all lemmata and theorems of Section 3 4 are also true for the extended system RS°°(with almost literally the same proofs)5, and as an easy consequence from Theorem 3.12 one obtains the

EMBEDDING THEOREM for KPM. If M G H and if H is closed under ξ *-> £R then for each theorem φ of KPM there is anneN such that H|$^ φM. Some more severe modifications have to be carried out on Section 4. The first part of this section (down to Lemma 4.5) has to be replaced by Section 1 of the present paper. Then the sets C(α,/3) are no longer closed under (ττ,£) i-> V>τr£ (£ < α)> but only under (ψ\α) as defined in Definition 1.2 above. Therefore we have to add "π,ξ G £*(£)" to the premise of Lemma 4.6c, and accordingly a minor modification as to be made in the proof of Lemma 4.7(^41). But this causes no problems. A little bit problematic is the fact that the function ^M is not weakly increasing. In order to overcome this difficulty we prove the following lemma.

DEFINITION 2.1. For 7 = uΛo H ----- h ω7n with 70 > Further we set e(0) :^= On.

> 7n we set 6(7) :=

LEMMA 2.1. For 7 G CM (7 + 1) and 0 < α < 6(7) the following holds *) ΨM(Ί + 1) < ΨM(Ί + «) & CM(Ί + 1) C CU(Ί + α) b)0• 3y G LM^.(^, 2/) L0 -> 3yGL M A(^,y) a* := 7, + ω^°^ < 7 + a;/*+ao+M =•a* < a. Lei 7Γ := ψMa* & ^ := ^Ma*. Then by 1.4.7 π G W-[θ] & TT < ^Mδ ίΓe o/so have WG?;(a* G CM(a*)) and thus ^ι^(βt < π). The Boundedness- Lemma gives us now W € T,( WS[θ][t] |f Γ, ii LO -> ByeL^ίί, y) ).

From this by an application of (/\) we obtain H~[Q\ \%- Γ, B(LV). From L.2.5H and L.3.10 we get W-[θ] || Γ, Ad(L*) with 8 := ω"+5. We also have WS[Θ] I-? Γ, Lπ i L 0 . 7/ence H-[θ] ^ Γ, L, g L0 Λ Λd(L π ) Λ app/y (V) arid obtain W- [θ] \^~ Γ ." (7) Replace I by M in the Corollary to Theorem 4.8 and in Theorem 4.9. This yields the following Theorem.

THEOREM. Let ϋ* := ψ^ (εM+l). Then for each Σl -sentence φ of £ we have: KPM h Mx(Ad(x] -> φ*) => L& \= φ.

COROLLARY.

|KPM| < Ψ

§3. Further properties of the functions φκ. We prove four theorems which together with L.1.3a,b,c and L.1.4a-e provide a complete basis for the definition of a primitive recursive well-ordering (OT,-